Pacific Journal of Mathematics

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1 Pacific Journal of Mathematics SIMPLIFYING TRIANGULATIONS OF S 3 Aleksandar Mijatović Volume 208 No. 2 February 2003

2 PACIFIC JOURNAL OF MATHEMATICS Vol. 208, No. 2, 2003 SIMPLIFYING TRIANGULATIONS OF S 3 Aleksandar Mijatović In this paper we describe a procedure to simplify any given triangulation of S 3 using Pachner moves. We obtain an explicit exponential-type bound on the number of Pachner moves needed for this process. This leads to a new recognition algorithm for the 3-sphere. 1. Introduction. It has been known for some time that any triangulation of a closed PL n-manifold can be transformed into any other triangulation of the same manifold by a finite sequence of moves [5]. We can describe the moves as follows. Definition. Let T be a triangulation of an n-manifold M. Suppose D is a combinatorial n-disc which is a subcomplex both of T and of the boundary of a standard (n + 1)-simplex n+1. A Pachner move consists of changing T by removing the subcomplex D and inserting n+1 int(d) (for n equals 3, see Figure 1). It is an immediate consequence of the definition that there are precisely (n + 1) possible Pachner moves in dimension n. We can now state Pachner s result [5] in the following way. Theorem 1.1 (Pachner). Closed PL n-manifolds M and N with triangulations T and K respectively, are piecewise linearly homeomorphic if and only if there exists a finite sequence of Pachner moves and simplicial isomorphisms taking the triangulation T into the triangulation K. In dimension 3 we have four moves from Figure 1 at our disposal. Using them, we can describe the main theorem of this paper. Theorem 1.2. Let T be a triangulation of a 3-sphere and let t be the number of tetrahedra in it. Then we can simplify the triangulation T to the canonical triangulation of S 3, by making less than a t 2 2 bt2 Pachner moves, where the constant a is bounded above by and the constant b is smaller than

3 292 ALEKSANDAR MIJATOVIĆ (2-3) (3-2) (1-4) (4-1) Figure 1. Three dimensional Pachner moves. The triangulation T in this theorem can be non-combinatorial (i.e., simplices are not uniquely determined by their vertices), as is the case with the canonical triangulation of S 3, consisting of two standard 3-simplices glued together via an identity on their boundaries. We should mention here that Pachner s original proof of Theorem 1.1 works for combinatorial triangulations only. However, at least in dimension 3, this does not matter because the second derived subdivision of any (possibly non-combinatorial) triangulation is always combinatorial and can be obtained from the original triangulation by a finite sequence of Pachner moves. A possible effect Pachner s result could have on the theory of 3-manifolds is discussed by the next proposition. Proposition 1.3. Let T and K be two triangulations of the same closed PL 3-manifold M. The existence of a computable function, depending only on the number of 3-simplices in T and K, bounding the number of Pachner moves required to transform T into K, is equivalent to an algorithmic solution of the recognition problem for M among all 3-manifolds. Proof. Assume first that f(t, k) is a computable function as described in the proposition. Suppose that T is a triangulation of M with t 3-simplices. Let K be a triangulation of some closed 3-manifold N containing k 3-simplices. Do all possible sequences of Pachner moves on the triangulation T of length at most f(t, k), and check each time if the result is isomorphic to K. This gives an algorithm to determine whether M and N are PL homeomorphic.

4 SIMPLIFYING TRIANGULATIONS OF S Conversely suppose that we have an algorithm to recognize M among all 3-manifolds. Now we need a complete (finite) list of all triangulations of all 3-manifolds with a fixed number of 3-simplices. In dimension three, such a list can be built algorithmically because there is an easy way of recognizing the 2-sphere (the Euler characteristic suffices) as a link of a vertex. We can now create all triangulations of M with the specific number of 3-simplices by running the recognition algorithm for M (which exists by assumption) on the list of all 3-manifold triangulations with the specified number of 3-simplices. An algorithm, making all possible Pachner moves on a triangulation of our 3-manifold M with t 3-simplices will after a finite number of steps (by Theorem 1.1) necessarily produce a given triangulation of M containing k 3-simplices. Since we can list all triangulations of M with t (respectively k) 3-simplices, this gives an algorithm to calculate the value of the function f(t, k) as required. At present there is no known algorithm to decide whether a given simplicial complex is an n-sphere, for n 4. This means that the proof of one of the implications in Proposition 1.3 breaks down in dimensions five and above since there is no way of building a list of all triangulations of all manifolds with a fixed number of top dimensional simplices, in these dimensions. The proof of the converse implication in Proposition 1.3, showing that a computable bound implies a recognition algorithm for a given n-manifold, remains valid in any dimension. Furthermore, if such a computable bound existed for all n-manifolds, and was independent of the underlying n-manifold, then it would give an algorithm to determine whether any two n-manifolds are homeomorphic. But using the fact (proved by A.A. Markov) that there is no such algorithm for n 4, we can conclude that such a computable function does not exist in dimensions four and above. It is interesting to note, that for any n-manifold M Pachner s theorem implies the existence of a function, depending only on the number of n- simplices in T and K, and bounding the number of Pachner moves necessary for the whole transformation. This is because there are only finitely many triangulations of our n-manifold M with fixed numbers of n-simplices. Then, using Theorem 1.1, a finite sequence of Pachner moves connecting any two of them, can be found. Taking the maximum length over this finite family of sequences gives us the bound. Therefore, computability of the function in Proposition 1.3 is an assumption that can not be omitted. The upper bound in Theorem 1.2 is computable. It therefore yields a new recognition algorithm for the 3-sphere.

5 294 ALEKSANDAR MIJATOVIĆ 2. Normal surfaces. One of the essential ingredients of the proof of Theorem 1.2 is the theory of normal and almost normal surfaces. In this section we shall describe some of its basic features. We will then go on to discuss the Rubinstein-Thompson algorithm [7] for recognizing the 3-sphere which provides the setting for the proof of Theorem 1.2. After it, we ll mention some of the consequences of normal surface theory which will prove to be useful later. At the end of this section we shall prove the isotoping lemma that will later give us a way of simplifying triangulations of the 3-sphere. Let s start with some definitions. A normal triangle (respectively quadrilateral) in a 3-simplex 3 is a properly embedded disc D, such that its boundary D intersects precisely three (respectively four) edges transversely in a single point and is disjoint from the remaining 1-simplices and vertices of 3. A normal disc is a normal triangle or quadrilateral. Figure 2. Three types of normal discs. There are four possible types of normal triangles, because each triangle is parallel to one of the faces of 3. Normal quadrilaterals will always separate the vertices of the tetrahedron in pairs. It is therefore clear, that we can only have three possible quadrilateral types. Together, there are 7 distinct normal disc types in a tetrahedron. Let M be a 3-manifold with a triangulation T. A properly embedded surface F in M is in normal form with respect to the triangulation T, if it intersects each tetrahedron of T in a finite (possibly empty) collection of disjoint normal discs. Since normal surfaces are always embedded, at most one of the quadrilateral types can occur in each 3-simplex. Suppose F is a normal surface in M with respect to T. Then F corresponds to a vector x = (x 1,..., x 7t ) with 7t coordinates, where t denotes the number of 3-simplices in the triangulation T. The index set {1,..., 7t} corresponds to all possible disc types in T (there is 7 of them for each tetrahedron). The coordinate x i is simply the number of copies of i-th disc type in our surface F.

6 SIMPLIFYING TRIANGULATIONS OF S Each 2-simplex in T contains three types of normal arcs (coming from normal discs), one cutting off each vertex of the triangle. If it is a face of two 3-simplices in T, then it gives rise to three matching (linear) equations, one corresponding to each normal arc type. Doing this for every triangle, not in the boundary of M, we ve constructed a linear system in 7t variables, consisting of at most 6t equations. It follows immediately from the construction, that the vector x, coming from the normal surface F, gives a solution to the linear system. By imposing extra conditions to ensure that all quadrilaterals in a given tetrahedron are of the same type, we obtain a restricted linear system. The conditions we ve just added are sometimes referred to as quadrilateral constraints. Now there is a one to one correspondence between embedded normal surfaces in M and nonnegative integral solutions to the restricted linear system. Haken proved that all nonnegative integral solutions to such a system are integer linear combinations of a finite set of nonnegative integral solutions x 1,..., x n, called fundamental solutions, which can be found in an algorithmic way. As it turns out, these fundamental solutions are characterized by the property of not having a decomposition as a sum of two (nontrivial) nonnegative integral solutions to the restricted linear system. Since each fundamental solution corresponds to an embedded normal surface, we obtain a finite set F 1,..., F n of embedded normal surfaces, called fundamental surfaces. Any embedded normal surface in M can thus be written algebraically as a nonnegative integer linear combination of fundamental surfaces. Miraculously, this algebraic fact carries over to geometry. In other words, we can define a geometric addition for any two normal surfaces F and G with the property that the sum of the corresponding solutions to the restricted linear system, is again a solution of the same system. This condition boils down to the fact that the union of all normal discs in both F and G satisfies the quadrilateral constraints. Assuming that and putting both surfaces in general position with respect to one another, cutting along the arcs of intersection in each tetrahedron, and pasting the pieces back together in the unique way, so that we end up with normal discs only, yields a well-defined embedded normal surface F + G. Its corresponding vector is a sum of the vectors coming from F and G. The cut and paste process described above is sometimes called regular alteration. An isotopy of the ambient manifold, preserving the normal structure of a given normal surface is called a normal isotopy. We should also note that the geometric addition described above is well-defined up to a normal isotopy of the summands. Before we describe the Rubinstein-Thompson algorithm, we need to introduce a concept, originally due to Rubinstein.

7 296 ALEKSANDAR MIJATOVIĆ Definition. A properly embedded surface in a 3-manifold M with a triangulation T is almost normal with respect to T, if it intersects each tetrahedron of T in a finite (possibly empty) collection of disjoint normal discs except in precisely one tetrahedron there is precisely one exceptional piece from Figure 3 and possibly some normal triangles. Figure 3. Almost normal pieces. This exceptional piece is either a disc (the first possibility in Figure 3) whose boundary is a normal curve of length eight (i.e., an octagon), or it is an annulus consisting of two normal disc types with a tube between them that is parallel to an edge of the 1-skeleton. Now we can describe the Rubinstein-Thompson algorithm which is designed to determine whether or not a 3-manifold M with a triangulation T is a 3-sphere. We can assume that M is closed, orientable and that H 1 (M; Z 2 ) is trivial. All these properties can be checked algorithmically. The last assumption guarantees that M contains no closed non-separating surfaces. The algorithm now is in three steps. We proceed as follows. Step 1. Find a maximal collection Σ of disjoint non-parallel normal 2-spheres in M. Step 2. Cut M open along Σ. This splits M into three different types of pieces: Type A: A 3-ball neighborhood of a vertex of T (every vertex is enclosed in such a piece). Type B: A piece with more than one boundary component. Type C: A piece with exactly one boundary component which is not of Type A.

8 SIMPLIFYING TRIANGULATIONS OF S Step 3. Search each Type C piece for an almost normal 2-sphere with an octagonal component. Conclusion: M is a 3-sphere if and only if every Type C piece contains an almost normal 2-sphere with an octagonal component. The bulk of the proof that this indeed is a recognition algorithm for the 3-sphere relies on the following two lemmas from [7]. Lemma 2.1. A Type B piece is a punctured 3-ball. Lemma 2.2. A Type C piece is a 3-ball if and only if it contains an octagonal almost normal 2-sphere. By Lemma 2.2, if some Type C piece fails to contain an octagonal almost normal 2-sphere, then it is not a 3-ball and M is not a 3-sphere. Otherwise, M is just a collection of 3-balls and punctured 3-balls glued together. Since every 2-sphere is separating, M has to be a 3-sphere. The difficult part of the argument is in the proof of Lemma 2.2. It is here that Thompson simplified Rubinstein s original methods to prove the existence of an octagonal almost normal 2-sphere in a 3-ball of Type C, by using Gabai s powerful notion of thin position. We should also note that the easier converse implication in Lemma 2.2 follows from Lemma 2.7. In order to be able find a maximal collection of disjoint non-parallel normal 2-spheres in M in an algorithmic way, we need the following lemma. Lemma 2.3. A maximal collection Σ of disjoint non-parallel normal 2- spheres in M, as in the Rubinstein-Thompson algorithm, can always be constructed algorithmically. A proof of this lemma was given by Casson [1]. It was also described in [3] (see Lemma 3). It will be important for us to be able to bound the complexity of all of the 2-spheres in the maximal collection Σ. We shall therefore give a brief description of this algorithm. Additivity of Euler characteristic implies at once that if there exists a nontrivial normal 2-sphere in our triangulation, we can also find one (which is also nontrivial) among fundamental surfaces. Since the family of fundamental surfaces is accessible in an algorithmic way, we can take this fundamental 2-sphere to be the first element in Σ. Assume now that we have already constructed a subcollection Σ of Σ. We shall look for normal surfaces with respect to the triangulation T, lying in the complement of the normal 2-spheres constructed so far. In any tetrahedron from T we can have complementary regions of Σ that are not of the form triangle I or square I (see Figure 4), as well as the ones that are. Note also that the unions of the product regions support a natural I-bundle structure. These I-bundles are usually referred to as parallelity regions. We can now describe normal surfaces in the complement of Σ by assigning a variable to each triangle or square type that does not lie in any of the

9 298 ALEKSANDAR MIJATOVIĆ parallelity regions. The equations are again the matching equations along the faces together with the equations that ensure that the surface intersects each parallelity region in a well-defined number of components. In other words each parallelity I-bundle contributes new linear equations (that are of the same form as the matching equations along faces) and no new variables. This is because the normal 2-spheres we are looking for, can only run parallel to the horizontal boundary in these I-bundles. Adding the usual quadrilateral constraints gives a restricted linear system. Like in standard normal surface theory, we have a one to one correspondence between closed normal surfaces in the complement of Σ and solutions of the above restricted linear system. Since addition of two solutions is again realized topologically by regular alteration, the same argument as before tells us that the next normal 2-sphere, that is not parallel to any of the elements in Σ, can be chosen from the family of fundamental surfaces. So in order to find a maximal family Σ of disjoint non-parallel normal 2- spheres, we just have to keep repeating this procedure. We stop when each normal 2-sphere in the complement of Σ is normally parallel to some normal 2-spheres in the collection Σ. Lemma 2.4 guarantees that this process has to reach such a stage. As far as the complexity, i.e., the number of normal pieces, of elements in Σ goes, at each stage it is going to be bounded by Proposition 2.5. Since the linear algebra in the proof of Proposition 2.5 (which can be found in [2]) depends only on the number of normal variables and is independent of how many equations we have, it is the number of different normal disc types outside the parallelity regions, that needs to be controlled. The proof of Lemma 2.4 shows that this number is bounded linearly by the number of tetrahedra in T. In fact there can be at most 11 different normal disc types in a single tetrahedron in T at any stage of the process. We shall calculate explicit upper bounds later on in this section. We still need to answer the question of how to search for octagonal almost normal 2-spheres that are contained in Type C pieces. Modified versions of standard normal surface theory algorithms suffice for the search. So our goal is to construct an algorithmic procedure which will find an octagonal almost normal 2-sphere in each Type C piece. These 2-spheres will exists by Lemma 2.2 if the 3-manifold M we are looking at is a 3-sphere. We proceed as follows. First we fix a tetrahedron H in the triangulation T of a 3-manifold M and then we fix a normal curve c of length eight on its boundary (there are three choices for c). Now an analogue to the normal surface theory, used to construct the collection Σ, can be set up. The matching conditions will look just like before. Quadrilateral constraints have to be modified however, because we want our solutions to consist of normal triangles and quadrilaterals everywhere except in H, where we want them to be composed

10 SIMPLIFYING TRIANGULATIONS OF S of normal triangles and octagonal components with boundaries parallel to c. The notion of regular alteration can be defined in this generalized setting and again it gives rise to the correspondence between integer linear combinations of the fundamental solutions to the (generalized) restricted linear system and the set of all surfaces described above. Fundamental surfaces are again the ones corresponding to fundamental solutions. We should also note that their complexity is bounded by Proposition 2.5 since the linear system they are the solutions of, has less than 11t variables. What we really want is to find algorithmically octagonal almost normal 2-spheres that are contained in Type C pieces. We know that one such 2-sphere exists in each Type C piece by Lemma 2.2. This 2-sphere can be expressed as a sum of the fundamental surfaces. Precisely one of the summands has to contain a single octagonal piece and, since the Euler characteristic is additive, at least one of the fundamental surfaces in the sum has to be a 2-sphere (since the Type C piece we are looking at contains an octagonal almost normal 2-sphere, it has to be a 3-ball and can therefore not contain embedded projective planes). If the fundamental 2-sphere in the sum does not contain an octagon, then it has to be normal and thus parallel to the unique normal 2-sphere from Σ that is bounding the Type C piece we are looking at. This is a contradiction because we could then isotope it away from all the other summands by a normal isotopy. Since regular alteration is defined up to normal isotopy, this would then make the sum (i.e., a 2-sphere) disconnected. So we ve found an octagonal almost normal 2-sphere in a Type C piece that is fundamental. The complexity of the fundamental octagonal almost normal 2-sphere we ve just constructed is bounded in the same way as all the other complexities of the normal 2-spheres in Σ. This follows directly from the construction, since all we are doing when searching for an almost normal 2-sphere, is just making another step of the recursion that gave us Σ, without increasing the number of normal variables. We will give an explicit estimate for the complexity later on in this section. Let s first bound the number of disjoint non-parallel normal 2-spheres in Σ. This is made possible by an old idea due to Kneser. Lemma 2.4. Let T be any triangulation of S 3 and let t be the number of tetrahedra in T. Then any family of disjoint non-parallel normal 2-spheres contains at most 6t of them. Proof. Normal triangles and squares chop up any tetrahedron in T into several pieces. But at most six of these regions are not of the form triangle I or square I (see Figure 4). Let n be the maximal number of disjoint non-parallel normal 2-spheres in T. Then the complement of this family has precisely (n + 1) components. Each of those components must contain at least one of the non-product

11 300 ALEKSANDAR MIJATOVIĆ Figure 4. Complementary regions which are not a product. regions. This is because any component, consisting only of product pieces, is bounded by two parallel normal 2-spheres. Since the total number of non-product regions is bounded by 6t, our lemma is proved. We are interested in bounding the number of normal pieces of elements in Σ. We also want to bound the number of normal pieces of the octagonal almost normal 2-spheres that arise in Type C pieces. Both of these things can be accomplished at one go, because we know that the procedure giving Σ can be extended (by making a single additional step) to an algorithm producing octagonal almost normal 2-spheres in Type C pieces. The proposition we are about to state is proved in [2]. It originally deals with the linear system in 7t variables coming from the matching equations for normal surfaces. Its proof uses some basic linear algebra on the linear system which consists of matching equations. We should note at this point that the number of equations in this linear system does not influence the bound that the proposition gives. Proposition 2.5. Let M be a triangulated 3-manifold containing t tetrahedra. Let x be a fundamental solution of a system of linear equations coming from matching conditions. Then each coordinate of the vector x is bounded above by 7t2 7t 1. Using Proposition 2.5, we can bound the size of each component of all the vectors corresponding to the normal 2-spheres in Σ. It follows from Figure 4 that the number of normal discs that are not contained in the parallelity regions (at any stage of the algorithm producing the family Σ) is always bounded above by 11t. The system of equations we are solving at each stage consists of the matching equations along the faces together with the equations that ensure that the surface intersects each parallelity region in a well-defined number of components. We should note here that the latter equations are of the same form as the matching equations.

12 SIMPLIFYING TRIANGULATIONS OF S Proposition 2.5 then implies that there can be at most 11t2 11t 1 parallel copies of a given normal disc type in a complement of any subcollection Σ from any stage of the algorithm. This can be deduced because the proof of Proposition 2.5 depends only on the number of variables and the shape of the equations, i.e., the number and size of nonzero coefficients in each equation. Since the number of variables has increased and the shape of the equations hasn t changed, we get the above bound by substituting 7t with 11t into Proposition 2.5. Lemma 2.4 tells us that we ll never have to make more than 6t steps when constructing Σ. This means that each of the normal disc types in the complement of any subcollection Σ can only give rise to less than (2 11t 2 11t 1 ) 6t normal discs of the same type in the initial triangulation T. This is because at each stage of the algorithm the number of parallel copies of a fixed normal disc type is given by 11t2 11t 1. We have to include the factor of 2 because in each parallelity region there are two normal variables contributing to the number of parallel copies of a given normal disc type in the initial triangulation. We can obtain a similar kind of bound for octagonal almost normal 2-spheres. We only have to change the exponent from 6t to (6t + 1). This is because all these octagonal almost normal 2-spheres are just one step away (in our algorithm) from the normal ones (bounding Type C pieces) and at each stage they are described by fewer variables. For example, in our original triangulation they require 7(t 1) + 4 variables. So the bound in Proposition 2.5 applies. Putting everything together and using the fact that 5t(11t2 11t ) 6t+1 < 2 110t2, we get the following lemma (the factor 5t comes in because there are at most 5 different normal and almost normal pieces in each tetrahedron of T ). Lemma 2.6. Let T be a triangulation of the 3-sphere which contains t tetrahedra. Then the number of all normal pieces contained both in all elements of Σ and in all octagonal almost normal 2-spheres from all Type C pieces is bounded above by 2 110t2. We should note at this point that this is the only part of the bound in Theorem 1.2 which contains a quadratic expression in its exponent. If one could find both the octagonal almost normal 2-spheres in Type C pieces and the maximal family Σ among the fundamental solutions of linear systems that are based on the triangulation T, the bound in Lemma 2.6 would have a linear function (similar to the one in Proposition 2.5) in its exponent. The essential process we are just about to describe, is the one of isotoping almost normal surfaces. It is going to provide a foundation for the simplifying procedure needed for the proof of Theorem 1.2.

13 302 ALEKSANDAR MIJATOVIĆ Let F be a separating almost normal surface in a 3-manifold M with a triangulation T. Its weight, w(f ), is defined to be the number of points in the intersection of F and the 1-skeleton T 1. If F contains an octagon, a natural isotopy is the one pushing the surface over an edge which meets the length eight normal curve bounding the octagon in two points. There are two possible natural isotopies, depending on the component of M F we are pushing into. In case of other non-normal pieces (see Figure 3), a natural isotopy pushes the tube part of the annulus so that it encompasses one of the edges it is parallel to. As a result in both cases, we get a surface with its weight equal to w(f ) 2. Notice that if we look at our almost normal surface F in the complement of the 1-skeleton of T, there is an obvious compression disc D for it, enveloping the edge we are isotoping over. The natural isotopy can then be realized by isotoping over the 3-ball bounded by D and the disc in F, bounded by D. The natural isotopy is only the first step in the process of isotoping almost normal surfaces. Everything else will be accomplished by a sequence of elementary isotopies. We can define them as follows. Let A be a 2-simplex in T, containing a non-normal arc (Figure 5) which comes from intersecting A with an isotope of F. A B e Figure 5. An isotope of F intersects A in a non-normal arc. A disc B in the triangle A (see Figure 5) is bounded by the non-normal arc and a subarc of the edge e. An elementary disc can be constructed by banding together two parallel copies of B in the complement of 1-skeleton, where the band runs around the edge e. Its boundary is a simple closed curve in the surface, bounding a disc on one side. An elementary isotopy is an isotopy over the 3-ball bounded by the disc in the isotope of F and the elementary disc we ve just defined. Since F is a separating surface, we can fix a complementary component I of M F. All the elementary isotopies that we are going to do from now on, are going to have the same direction. We will always be isotoping towards the interior of the component I. The following isotoping lemma will play a crucial role in the simplifying process. A similar result is proved in [6] by a careful inspection of all the possible cases. The proof we are giving here is based on elementary isotopies

14 SIMPLIFYING TRIANGULATIONS OF S and is better suited from our perspective because it sheds more light on the side of things we ll be interested in later. Lemma 2.7. Let F be a separating almost normal surface in a 3-manifold M with a triangulation T. Let I be a component of the complement M F, if F contains an octagon. Otherwise let I denote the component containing a solid torus region in the interior of the 3-simplex where F is not normal. A natural isotopy followed by a sequence of all possible elementary isotopies, all going in the direction of I, will result with a surface intersecting each tetrahedron of T in pieces as in Figure 6 and in normal pieces. Moreover, in each tetrahedron there can only be at most one piece of the first type from Figure 6. A single 3-simplex can contain several pieces of all the other types in Figure 6 as well as several normal pieces. The pieces in Figure 6 can not be parallel. Figure 6. Non-normal pieces in the tetrahedra of T. Proof. First note that after the natural isotopy, all the non-normal arcs we get will give rise to elementary isotopies in the direction of I. After each isotopy both F and I will change, but we ll still denote both resulting spaces by F and I respectively. After the natural isotopy, F and I satisfy the following conditions: 1) In each tetrahedron of T the component I consists of a family of 3- balls, each one bounded by pieces of F and a (possibly disconnected) planar surface, contained in the boundary of the 3-simplex. 2) Each 3-ball from 1 intersects any face of the tetrahedron it lies in, in at most one disc.

15 304 ALEKSANDAR MIJATOVIĆ An elementary isotopy moves a disc in F over a 3-ball in I, which intersects a single edge e in T 1. So the new I is just the old I without the 3-ball we isotoped over. This 3-ball is a union of a family of 3-balls, one in each tetrahedron of the star of e. In fact there can be more than one 3-ball from the same family in a single 3-simplex if this 3-simplex occurs more than once in the star of the edge e. This is perfectly feasible in a non-combinatorial triangulation, but it does not have any effect on the process we are studying. The elements of the above family are the ones that are going to determine the topology of the pieces of I in tetrahedra of T. In fact, each 3-ball from condition 1 will after an elementary isotopy still satisfy both conditions if we substitute the old F with the new one. So after performing all possible elementary isotopies towards the interior of I, the surface F we end up with will intersect each triangle of T in normal arcs and simple closed curves which miss the boundary of the triangle. I I I I I I Figure 7. An intermediate state of the isotopy on a triangle in T. The region I will, after the isotopy, consist of 3-balls in each 3-simplex. There is going to be a bijective correspondence between the 3-balls in the end, and the ones we started with. By condition 2, every 3-ball will still intersect any face of the 3-simplex it lies in, in at most one disc. It is also true that the number of these discs will not increase when we pass from the 3-ball pieces of I at the beginning to the 3-balls at the end. Let s look at the pieces of F in each tetrahedron. It is obvious that all the possibilities of the lemma can actually arise. We have to see that they are the only ones. Claim. A single piece of F can intersect a triangle of T in either a unique normal arc or in a single simple closed curve. Every piece of F is contained in the boundary of a 3-ball piece of I. This 3-ball intersects each triangle of the 2-skeleton T 2 in at most one disc. So no triangle can contain two simple closed curves or a simple closed curve and a normal arc, both belonging to the same piece of F. The same argument tells us that a triangle in T 2 can either contain two normal arcs of intersection with a single piece of F or at most three of them, each one cutting off a vertex of the triangle in the 2-skeleton.

16 SIMPLIFYING TRIANGULATIONS OF S Now we need to prove that our piece of F can have at most one normal simple closed curve boundary component. So assume the opposite. Since the piece is a subset of the boundary of a 3-ball, no arc contained in it, running between two distinct boundary components of our piece, can be extended to a simple closed curve in the 2-sphere bounding that 3-ball, without increasing the number of intersection points with the boundary of our piece. On the other hand, assuming we have at least two normal simple closed curves in the boundary, there surely exist two normal arcs, belonging to the distinct boundary components of our piece, that are contained in a single 2-simplex. Connecting them by an arc in the piece of F contradicts what was said before (because these two normal arcs are both contained in the boundary of a disc in the 2-simplex they lie in). So now it follows that the piece of F we are looking at, can contain at most one normal boundary component which is of length at most eight. This is because the only normal curve of length 12, intersecting each 2-simplex (in the boundary of a tetrahedron) in 3 normal arcs as above, consists of 4 simple closed curves (one for each vertex of the tetrahedron). It is also well-known that normal simple closed curves of lengths 9, 10 or 11 do not exist. There are precisely three normal simple closed curves of length eight in the boundary of a tetrahedron. So if our piece of F is bounded by one such curve, then at least one of the faces of the tetrahedron intersects the 3-ball piece of I (containing in its boundary the piece of F we are considering) in two discs. This is a contradiction that proves the claim. The claim implies the following seven possible boundaries for any piece of F : Normal simple closed curve of length three, normal simple closed curve of length four, single simple closed curve, normal simple closed curve of length three and a simple closed curve, two simple closed curves, three simple closed curves, four simple closed curves. Since every piece of F in any tetrahedron is planar, it is up to homeomorphism determined by its boundary. This implies that all possible pieces of F are the ones listed in the lemma. The fact that all these planar surfaces are embedded as in Figure 6 (up to an isomorphism of the tetrahedron) follows from the observation that all the elementary discs are parallel to edges of the 1-skeleton. 3. Outline of the proof. Given a triangulation T of the 3-sphere, how do we simplify it? The process is divided into two stages. First, we create a subdivision S of T by defining it in each complementary piece of the manifold S 3 Σ in such a way that the triangulations match along all normal 2-spheres in Σ. The second step consists of simplifying S down to the canonical triangulation of S 3.

17 306 ALEKSANDAR MIJATOVIĆ An explicit construction of S, using Pachner moves, will be given in Section 5. The simplifying procedure of step two is based on the relationship between Pachner moves and shellable triangulations. This relationship will be established in Section 4. Now, we are going to describe the additional structure on the complementary pieces of S 3 Σ, needed for the definition of the subdivision S. We already know (Lemma 2.2) that every Type C piece contains an octagonal almost normal 2-sphere. To see that each Type B piece also contains an almost normal 2-sphere, it is useful to introduce an ordering on the normal family Σ. It comes naturally by picking a vertex of T and looking at the complementary region (which is not a Type A piece) of the trivial normal 2-sphere around it. Topologically we get a 3-ball containing our normal family Σ. Now, the ordering on Σ is induced by inclusion. For example, the trivial normal 2-sphere around the vertex we removed is the unique largest element. The smallest elements in this ordering are either trivial normal 2-spheres consisting of normal triangles only or the ones bounding Type C 3-balls. Our task is to find an almost normal 2-sphere in a piece with more than one boundary component. Pick the largest 2-sphere in its boundary. A very nice argument in [7] (subclaim ) implies that there must be an edge in T with a subarc which runs from the largest component of the boundary to some other component and whose interior is disjoint from Σ. By taking parallel copies of the two 2-spheres connected by this arc in the piece we are looking at, and tubing them together in one of the tetrahedra in the star of the edge, we obtain our almost normal 2-sphere. All the almost normal 2-spheres we ve created are separating, because we are in S 3. By picking the right complementary component in S 3 and applying Lemma 2.7, we can simplify each almost normal 2-sphere by a sequence of elementary isotopies. Since we are only using elementary isotopies going in the same direction (towards the interior of a fixed complementary component in S 3 ), the whole process can be realized by an embedding of S 2 I, where the top 2-sphere is the almost normal 2-sphere we started with and the bottom one is the 2-sphere coming from Lemma 2.7 (see Figures 8 and 9). Another important point here is that the whole isotopy never leaves the Type B (or C) piece it started in. This is true simply because an analogous statement holds for each elementary isotopy. This implies that the isotoped surface coming from Lemma 2.7 will be contained in the interior of the piece containing the almost normal 2-sphere we started with. In a Type C piece, the isotopy can go in two directions because the almost normal 2-sphere in this case contains an octagon. The surface we get, when isotoping towards the interior of the piece, will have 0 weight. This follows from the observation that we can forget about all pieces in Figure 6 if we

18 SIMPLIFYING TRIANGULATIONS OF S compress each annulus with a length three normal curve in its boundary. This would then give a family of normal 2-spheres contained in the Type C piece which is a contradiction. Therefore, the 2-sphere we end up with has to miss the 1-skeleton. Similar reasoning tells us that an isotopy in the other direction in the Type C piece has to end with a 2-sphere, intersecting the 2-skeleton T 2 in normal curves parallel to the ones coming from the boundary of the piece we are looking at and possibly in some simple closed curves which miss the 1-skeleton. The almost normal 2-spheres in Type B pieces that we are going to consider, will never contain an octagon. We will therefore be isotoping in one direction only. Using the same kind of arguments as before, we can conclude that the 2-sphere we end up with consists of boundary components of the Type B piece (all except the two we started with, see Figure 9), tubed together by pieces depicted in Figure 6. It should be noted that if a Type B piece has only two boundary components, then the isotoped 2-sphere does not intersect the 1-skeleton. Let Λ be the following collection of 2-spheres: In every Type C piece just take an octagonal almost normal 2-sphere which exists by Lemma 2.2. In each Type B piece take a copy of the almost normal 2-sphere described above with the annulus connecting two normal pieces moved by a natural isotopy, so that it envelops the edge it is parallel to. The 2-spheres from Λ in Type B pieces are therefore normal in all the tetrahedra of T, except in the ones contained in the star of the edge we isotoped over. The sequences of elementary 3-balls, corresponding to the supports of elementary isotopies, yields the additional structure (on Type B and C pieces) that is required to define the subdivision S. Elementary discs and an element in Λ chop up each Type C piece of S 3 Σ (see Figure 8). In the case of a Type B piece, the element of Λ will, after the isotopy, consist of all but two boundary normal 2-spheres tubed together by pieces described in Figure 6. Again, the Type B piece in question can be decomposed into (many) 3-balls and two punctured 3-balls. The two punctured 3-balls come from an element in Λ we started our isotopy on, and from its isotope after we ve performed all elementary isotopies on the tubed almost normal 2-sphere (see Figure 9). The rest of the Type B piece is decomposed into 3-balls by all the elementary discs required for this isotopy. After the isotopy from Lemma 2.7, what s left in each tetrahedron of the complementary component we isotoped into, are just 3-balls bounded by the pieces from Lemma 2.7 on one side and possibly some normal pieces of the elements in Σ on the other. Schematically, the situation after the isotopy is depicted by Figure 9. Now we want to triangulate all of these 3-balls (the elementary ones as well as the ones that are left over in the component we were isotoping into),

19 y s 308 ALEKSANDAR MIJATOVIĆ i s o to p y p o o t i 3-ball X Figure 8. Two sequences of elementary isotopies in a Type C piece. C C B A Figure 9. The tubed almost normal 2-sphere and the isotoped 2-sphere in a Type B piece. in all the pieces of the complement of Σ, by simple shellable triangulations. Since all the processes described above induce polyhedral structures in the boundaries of all the 3-balls (this will be described in detail in Section 5) in question, subdividing the boundary 2-spheres in order to obtain genuine triangulations and then coning them, does the job. Doing so in every piece of the space S 3 Σ exhausts the whole 3-sphere and therefore completely determines the subdivision S. The fact that all these cones are indeed shellable, is proved in [4] (Lemma 5.4). Here we are relying on the property that all the bounding 2-spheres we ll need to cone in the process, are triangulated by combinatorial triangulations. The reason why we want these 3-balls to be shellable is simply

20 SIMPLIFYING TRIANGULATIONS OF S because each elementary isotopy can then be realized by a shelling of the corresponding 3-ball. So what we really want from the cones on the 2-spheres above, is to be shellable without ever having to shell from the faces contained in a fixed disc, which lies in the bounding 2-sphere. This disc is just a 2-manifold along which the 3-ball, we are trying to triangulate, is glued onto the rest of the (Type B or C) piece. This can always be achieved since a cone on a disc with a combinatorial triangulation can be shelled from the side just by coning the shelling procedure of the disc itself. The simplifying process works its way up the ordering of the normal 2- spheres in Σ. First we change the subdivision S in each Type C piece (which are smallest elements in our ordering), making it a cone on the unique boundary component. In Section 4 we will discuss how to implement elementary shellings from a 2-sphere boundary component by Pachner moves, if on the other side of that 2-sphere we have a cone on it. Using that construction we can pick a 3-ball piece (in some 3-simplex), contained in the 3-ball X from Figure 8, and turn the whole 3-ball X, bounded by the 2-sphere coming out of Lemma 2.7, into a cone on its boundary. This is simply because the complement (in X) of the coned 3-ball piece we picked, is shellable. That follows from the observation that all the (coned) 3-ball regions from Figure 6, the 3-ball X is made of (we already know that the first possibility in Figure 6 can not occur) can be viewed as vertices of a graph whose edges correspond to the discs in the interiors of the 2-simplices of T. Since this graph is a tree (this follows from the fact that the isotoped 2-sphere bounds a 3-ball), there is a global shelling strategy for the complement of the piece we picked in the 3-ball X. This can be made simplicial by shelling one cone at a time. So now we can assume that the 3-ball X is coned. We can carry on by shelling (in the reversed order) all elementary 3-balls (and the 3-ball corresponding to the natural isotopy) involved in the isotopy taking the almost normal 2-sphere Y to the boundary of X. By this stage, we ve changed the subdivision S so that it looks like a cone on Y in the Type C piece we are looking at. Above it, S is still unchanged. We can now do the same thing towards the boundary of the piece we are considering, again using the shellable nature of the subdivision S in all appropriate 3-balls. What we have now is a cone on the isotope of Y. Now we concentrate on the S 2 I region between the isotope of the octagonal almost normal 2-sphere Y and the single boundary component of the Type C piece. We first shell all the 3-balls from that region that are bounded by pieces in Figure 6. We thus obtain a cone on a normal 2-sphere which is parallel to the bounding 2-sphere of our piece. Since all the regions between any two parallel normal pieces are cones as well, we can shell them one by one and therefore get a cone on the boundary of our piece. Here we are relying on the fact that the normal structure on the bounding 2-sphere is shellable. In

21 310 ALEKSANDAR MIJATOVIĆ general this needn t hold, but the technical assumption that we are going to make on our triangulation T at the beginning of Section 5 will guarantee this property. This completes the simplification of the triangulation S in all Type C pieces. Take a Type B piece and assume that all normal 2-spheres, strictly smaller than the largest normal 2-sphere in its boundary, already bound coned 3- balls. The strategy now is similar to the one we used in Type C pieces. Since the tube of the 2-sphere element of Λ in our piece runs from the largest boundary component to some other boundary component, we can deduce that all other normal 2-spheres in the boundary that are going to be tubed together by the isotoped 2-sphere (see Figure 9), are going to bound cones on one side. We first shell all the regions which are bounded by two parallel normal pieces and lie between a normal boundary component and the isotoped 2- sphere. We can do that by expanding the cone structures on the other side of the boundary components of our Type B piece, that exist by assumption. Now the 3-ball bounded by the isotoped 2-sphere from Λ is again chopped up into 3-ball regions that are glued together along discs contained in the interiors of 2-simplices of T. Like before, there is a sequence of elementary shellings which gives a way of changing the triangulation of the 3-ball bounded by the isotoped 2-sphere from Λ to the cone on its boundary. We will now mimic what we did in Type C pieces. Let s take the sequence (in the reversed order) of all elementary 3-balls coming from the elementary isotopies needed to push the 2-sphere element of Λ in our Type B piece, down to the 2-sphere which now already bounds a coned 3-ball. Using this sequence in the same way as above, we can change the triangulation S in our piece to the cone on the 2-sphere element of Λ. It is now obvious how to simplify the remains of the subdivision S in the Type B piece we are looking at. So we ve managed to transform the subdivision in our piece into a cone on the largest boundary component. Now we want to make sure that the techniques described above suffice for the total simplification of the subdivision S. Let s assume that K is a triangulation of S 3 and that S is its subdivision containing Σ as a subcomplex. Let x be the vertex of K inducing an ordering on Σ. Let s also assume the following property: If all elements of Σ, smaller then a given normal 2-sphere A in Σ, bound coned 3-balls, then using Pachner moves we can change the triangulation of the 3-ball bounded by A into a cone on A, without altering the simplicial structure of A. Then we claim that we can transform S, using Pachner moves only, into a cone on x glued to another copy of itself via an identity on the boundary. To see this, we ll use a simple induction on the depth of elements in Σ. A normal 2-sphere in Σ is of depth k if it is greater than precisely k elements of Σ.

22 SIMPLIFYING TRIANGULATIONS OF S We can use our assumption for the 2-spheres of depth 0. Let A in Σ be of depth (k + 1) and assume we ve coned all the 2-spheres of depth smaller or equal to k. Any 2-sphere smaller than A is of depth at most k. So we can use the assumption again. This proves our claim and therefore completes the simplification process. 4. Pachner moves and shellable triangulations. In this section we are going to establish a relationship between elementary shellings and Pachner moves. We will do this in dimensions two and three. Both cases will play a crucial role in building and simplifying the subdivided triangulation S. Let s start by stating precisely what we mean by shelling. Definition. Suppose that M is a submanifold of a triangulated n-manifold M with boundary. If there exists an n-simplex in the triangulation of M with the property that M is a combinatorial (n 1)-disc, such that M equals the closure (in M) of the complement M, then we say that M is obtained from M by an elementary shelling. An elementary shelling is quite similar to an elementary collapse of the top dimensional simplex. The crucial difference lies in the fact that here we stipulate explicitly that the resulting space has to be a manifold. Another thing which is worth mentioning is that the boundaries M and M differ by a single n 1 dimensional Pachner move. A sequence of such elementary shellings is called a shelling. Saying that a triangulation of an n-manifold is shellable simply means that there exists a sequence of elementary shellings which will reduce the triangulation down to a single n-simplex. Since the homeomorphism type of the manifold in question does not change under an elementary shelling, it is clear that n-balls are the only candidates to have shellable triangulations. It is for example very well-known that any combinatorial triangulation of the two dimensional disc is always shellable. As it was mentioned before, Lemma 5.4 in [4] and the above observation about discs together imply that a cone on any combinatorial triangulation of the 2-sphere constitutes a shellable triangulation of the 3-ball. Now we are going to express all possible elementary shellings by Pachner moves in the following three dimensional situation. Suppose we had a triangulated 3-manifold and we wanted to make an elementary shelling from a 2-sphere boundary component. Suppose further that on the other side of this 2-sphere, we had a cone on it. We have to consider three different cases according to the number of faces of the 3-simplex we are shelling, which are contained in the boundary 2-sphere. The first case, where we have a single triangle in the boundary, is dealt with by Figure 10.

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